Nash strategies for the inverse inclusion Cauchy-Stokes problem

Abstract

We introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms is not straightforward. We reformulate the problem as a three-player Nash game. Thanks to an identifiability result derived for the Cauchy-Stokes inclusion problem, it is enough to set up two Stokes boundary value problems, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. We used a level-set approach to get rid of the tricky control dependence of functional spaces, and we provided the third player with the level-set function as strategy, with a cost functional of Kohn-Vogelius type. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for three different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems.

A Nash-game approach to joint data completion and location of small inclusions in Stokes flow

“…If we consider the case where possible object locations are known, concentrating only on the data completion problem, then the existence of a two-players Nash equilibrium can be proved by using the partial ellipticity of J 1 and J 2 with respect to η and τ , respectively [7,8]. This property allows us to restrict the search for Nash equilibria to bounded subsets of the strategy spaces, which remains consistent with the classical results of conditional stability of the Cauchy problem [1].…”

“…Then, the inverse problem under consideration here consists of a coupled inverse problem of object detection and data completion. The identifiability result for the inverse inclusion Cauchy-Stokes problem, with a homogeneous Neumann condition imposed on the unknown geometry, of Habbal et al [8] implies that a single pair of compatible-measurements is enough to recover the unknown objects and the missing boundary data on the remaining -inaccessible-part of the exterior boundary. It is well known that the considered inverse problems are ill-posed, in Hadamard's sense [9].…”

“…Illposedness makes classical numerical methods usually inappropriate, and consequently, some regularization techniques have to be used to solve the problem numerically. In order to address the present coupled ill-posedness, we use a game-theoretic framework, following the works [7,8]. We then suggest an alternating minimization approach, where the coupled inverse problem is formulated as a three players non-cooperative Nash game.…”

A three-player Nash game for point-wise source identification in Cauchy–Stokes problems

“…In step II, in order to solve the problems of partial optimization of J 1 , J 2 and J 3 , we need to calculate the gradient of these costs with respect to their respective strategies η, τ and λ. The fast computation of the latter is classical, and led by means of an adjoint state method, as shown by the proposition 2 and 3, the proof of the proposition 3 is given in ( [16], Appendix A. 1).…”

“…The same approach has been investigated in [6] for the solution of coupled conductivity identification and data completion in cardiac electrophysiology, and in [16] to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid using partial boundary measurements.…”